System with sub-nyquist signal acquisition and transmission and associated methods

ABSTRACT

A sensing and recovery system includes a sensing unit and a recovery unit coupled together. The sensing unit includes a sensor to generate a bandlimited continuous time analog signal, and a modulator coupled to the sensor to generate a modulated analog signal based upon modulation of the bandlimited continuous time analog signal at a modulating rate at least equal to a Nyquist rate for the bandlimited continuous time analog signal. A compressive sensing circuit is coupled to the modulator to generate a compressed sensed signal based upon conversion of the modulated analog signal at a sampling rate less than the Nyquist rate. The recovery unit recovers the bandlimited continuous time analog signal from the compressed sensed signal.

FIELD OF THE INVENTION

The present invention relates to the field of data communications, andmore particularly, to a sensing and recovery system and method usingcompressive sensing.

BACKGROUND OF THE INVENTION

In digital signal analysis, a signal is typically reconstructed fromdiscrete measurements. Typical approaches to sampling signals or imagesfollow Shannon's theorem wherein a sufficient sampling rate is at leasttwice the bandwidth (i.e., a lowpass or bandpass bandwidth depending onthe signal type) present in the signal. This minimal sampling rate isknown as the Nyquist rate or frequency.

This principle underlies nearly all signal acquisition protocols used inconsumer audio and video electronics, medical imaging devices, radioreceivers, etc. In reference to data conversions, for example, astandard analog-to-digital converter (ADC) implements the quantizedShannon representation in that the signal is uniformly sampled at orabove the Nyquist rate.

Even though traditional approaches to data acquisition depend on aNyquist rate, in many applications the Nyquist rate far exceeds thenecessary sample rate needed (or the number of samples over the timerecord) to accurately reconstruct the signal. For efficient storage orcommunications of the signal information content, data compression is animportant processing step prior to storage or transmission.

However, there are penalties associated with implementing datacompression. One such penalty is that the original information bearingsignal will likely be sampled above the Nyquist rate, and thus producemany samples with redundant information content about the signal. Thisprocess can at times require an ADC with a very high sample rate. Highsample rate converters tend to require higher power and are oftenlimited in resolution with respect to their lower sample ratecounterparts. This induces a system penalty of low resolution and highpower. In some cases it is difficult to even purchase or construct anADC with the required sample rate given the Nyquist viewpoint of thesignal.

Another penalty is induced by the data compression step, as performedwith a Karhunen-Loève transform, for example. The data compression stepattempts to remove the information redundancy in the samples, induced bythe Nyquist view of the conversion, so that a minimal set of samplesresult to sufficiently represent the information content of the originalsignal. Implementing compression prior to storage or transmissionrequires some type of computing resource, which in turn increases thepower drawn by the sensor system. The larger the oversampling factor ina Nyquist paradigm, the more severe these penalties become.

Recent developments have shown that compressive sampling or compressivesensing can provide sub-Nyquist rate sampling for communicationssystems. One such approach is disclosed in U.S. Patent No. 2011/0090394to Tian et al. A disclosed method of signal processing includesreceiving at a processor a data packet comprising compressively sensedor measured data of a signal, with the compressively measured datacomprising wavelet transform coefficients. The received signal is adiscrete signal, which in turn, requires transform coding before beingprocessed by the processor. The processor reconstructs the signal usinga clustering property of the wavelet transform coefficients. Adisadvantage of this approach is that transform coding of the discretesignal before being compressively sampled requires additionalprocessing, which in turn consumes power. Additional power consumptionmay be undesirable, particularly for battery-powered systems.

SUMMARY OF THE INVENTION

In view of the foregoing background, it is therefore an object of thepresent invention to provide a system that efficiently processes signalsusing compressive sensing for a reduction in overall size, weight andpower relative to traditional signal acquisition approaches.

This and other objects, features, and advantages in accordance with thepresent invention are provided by a sensing and recovery systemcomprising a sensing unit including a sensor configured to generate abandlimited continuous time analog signal, and a modulator coupled tothe sensor and configured to generate a modulated analog signal basedupon modulation of the bandlimited continuous time analog signal at amodulating rate at least equal to a Nyquist rate for the bandlimitedcontinuous time analog signal. A compressive sensing circuit is coupledto the modulator and is configured to generate a digital signal basedupon conversion of the modulated analog signal at a sampling rate lessthan the Nyquist rate of the original source signal. A recovery unit iscoupled to the sensing unit and is configured to recover data samplesthat would have resulted from Nyquist sampling of the original sourcesignal. The recovered discrete-time samples are then converted to abandlimited continuous time analog signal.

The sensor advantageously senses the continuous time analog data in itsinherent or indigenous domain. The input signal, once modulated, maythen be passed directly to the compressive sensing circuit without thesensing unit having to perform any sort of transform coding on thesignal before being compressively sampled.

The compressive sensing circuit may comprise an analog-to-digitalconverter or digitizer, for example. Also, the sample rate in thecompressive sensing circuit is substantially reduced from Nyquist-basedparadigms. Because the sensing unit need not oversample and compress thebandlimited continuous time analog signal, processing steps typical ofNyquist-based data acquisition are avoided, thus resulting in a savingsof size, weight and power of the sensor, as well as, a reduction instorage or communication requirements for the sensed data.

The sensing unit may further comprise a forward error correction (FEC)module configured to add error correction symbols to the compressedsensed signal. Similarly, the receiving unit may further comprise anerror detection and correction module configured to correct for errorsin the compressed sensed signal based on the error correction symbolsadded by said FEC module.

The sensing unit may further comprise a data integrity module configuredto add authentication symbols to the compressed sensed signal.Similarly, the receiving unit may further comprise an integrity checkmodule configured to authenticate the sensing unit based on theauthentication symbols added by the data integrity module.

Another aspect is directed to a method for sensing data comprisinggenerating a bandlimited continuous time analog signal, and generating amodulated analog signal based upon modulation of the bandlimitedcontinuous time analog signal at a modulating rate at least equal to aNyquist rate for the bandlimited continuous time analog signal. Acompressed sensed signal is generated upon conversion of the modulatedanalog signal at a sampling rate less than the Nyquist rate. The methodmay further comprise transmitting the compressed sensed signal to arecovery unit.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a sensing and recovery system in accordancewith the present invention.

FIG. 2 is a more detailed block diagram of the sensing unit shown inFIG. 1.

FIG. 3 is a more detailed block diagram of the recovery unit shown inFIG. 1.

FIG. 4 is a flowchart illustrating a method for sensing data inaccordance with the present invention.

FIG. 5 is a block diagram of a data handling system interfacing with aremote data storage facility in accordance with the present invention.

FIG. 6 is a more detailed block diagram of the data handling systemshown in FIG. 5.

FIG. 7 is a flowchart illustrating a method for operating a datahandling system in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention will now be described more fully hereinafter withreference to the accompanying drawings, in which preferred embodimentsof the invention are shown. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein. Rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art.

Compressive sensing enables a high resolution capture of physicalsignals from relatively few measurements, which may be well below thenumber expected from the requirements of the Shannon/Nyquist samplingtheorem. Compressive sensing makes use of a sparse representation forthe received signal, and then measurements are used to extract themaximum amount of information from the sparse representation for thereceived signal using a minimum amount of measurements.

Compressive sensing as discussed herein is initially focused on theprocessing of continuous time and continuous amplitude bandlimitedwaveforms, commonly referred to as analog signals. More specifically,the subject signals of interest are analog signals that have a sparserepresentation in an analysis domain.

As will be explained in greater detail below, signal acquisition andtransmission of analog signals using compressive sensing allows for anumber of benefits, such as power reduction for a battery-poweredsystem, a system that is signal agnostic when acquiring signals, asystem that has a low probability of intercept when transmittingsignals, a system that can recover the signals without aliasing errors,and a system that is resistant to spoofing.

Referring initially to FIG. 1, a sensing and recovery system 10 includesa sensing unit 20 and a recovery unit 60. The sensing unit 20 includes asensor 22 to generate a bandlimited continuous time analog signal. Amodulator 24 is coupled to the sensor to generate a modulated analogsignal based upon modulation of the bandlimited continuous time analogsignal at a modulating rate at least equal to a Nyquist rate for thebandlimited continuous time analog signal.

A compressive sensing circuit 30 is coupled to the modulator 24 togenerate a compressed sensed signal based upon conversion of themodulated analog signal at a sampling rate less than the Nyquist rate.The recovery unit 60 is coupled to the sensing unit 20, and includes asignal reconstruction circuit 62 to recover the bandlimited continuoustime analog signal from the compressed sensed signal. The circuitry ofthe sensing unit 20 and the recovery unit 60 are illustratively poweredby their respective batteries 27, 67.

A transmitter 70 is coupled to the sensing unit 20, and a receiver 80 iscoupled to the recovery unit 60. The compressed sensed signal generatedby the sensing unit 20 is provided to the transmitter 70 fortransmission via an antenna 72 to the receiver 80. The transmittedsignal is received by the receiver 80 via an antenna 82, which thenprovides the compressed sensed signal to the recovery unit 60.

In the illustrated embodiment, the interface between the sensing andrecovery system 10 is a wireless interface. Transmission of thecompressed sensed signal is not limited to any particular frequency ortransmission scheme, as readily appreciated by those skilled in the art.Alternatively, the interface between the sensing unit 20 and therecovery 60 may be a wired interface. Other configurations include thetransmitter 70 and receiver 80 configured as modems, for example.

The sensing unit 20 and the sensor 22 are signal agnostic, which meansthat the sensor is not customized for any specific signal type. Thebandlimited continuous time analog signal can be any analog signal thatis also continuous in amplitude. The analog signal can exist in anydomain that is naturally occurring, such as in the frequency domain,time domain, audio, pressure, and even images, for example. Essentially,the bandlimited continuous time analog signal can be based on anyphysically sensed variable.

In the healthcare field, the sensing unit 20 may be used to monitor apatient's heart rhythm, for example. Another application is with respectto spectral monitoring. In spectral monitoring, the sensing unit 20looks for narrowband signals in a large surveillance bandwidth. Thebandwidth may be in the GHz range, for example. The signals areadvantageously sampled at a rate less than the Nyquist rate, yet therecovery unit 60 is able to recover the signals without aliasing.

A more detailed block diagram of the sensing unit 20 will now bediscussed in reference to FIG. 2. The sensor 22 senses analog data andprovides a bandlimited continuous time analog signal, represented asx(t), to the modulator 24. The signal x(t) is in its inherent orindigenous domain.

The illustrated modulator 24 is a bi-phase modulator that multiplies thebandlimited continuous time analog signal by +1 and −1 to generate amodulated analog signal. The purpose of the modulator 24 is to spreadthe frequency content of the input signal so that the informationcontent is not destroyed by the low pass filter 26. The bi-phasemodulator 24 has a bi-phase modulation rate that meets or exceeds theNyquist rate of the sensed signal. The bi-phase modulation may also berandomly generated. As readily appreciated by those skilled in the art,the modulator 24 is not limited to a bi-phase modulator, and there existmany methods of generating a random bi-phase signal. However, therecovery unit 60 needs to know the exact sequence of the modulation inorder to properly decode the transmitted data. Consequently, the sensingunit 20 and the recovery unit 60 are synchronized.

The modulated analog signal is passed through an analog low pass filter26. The analog filter 26 has a filter response of h(t), which may bearbitrary but is known by the recovery unit 60. The filtered modulatedanalog signal is passed directly to the compressive sensing circuit 30.

The combination of the modulator 24, the analog filter 26 and thecompressive sensing circuit 30 provides a sufficient number of samplesto reconstruct the original signal, but at a sample rate below theShannon/Nyquist predictions. The processing steps and power consumptionof typical Nyquist theory data acquisition systems are avoided, which,in turn, prolongs the life of the battery 27. Since the sensing unit 20directly acquires samples in a so-called compressed domain, the explicitstep of data compression, and its impact on sample rate, is completelyavoided. This advantageously results in a power savings and simplercircuit realizations in the sensing unit 20.

The compressive sensing circuit 30 may be configured as ananalog-to-digital converter (ADC) or digitizer, as readily appreciatedby those skilled in the art. The compressive sensing circuit 30 operatesat a sampling rate less than the Nyquist rate on the filtered modulatedanalog signal. The sample rate may be represented as T_(S) _(—) _(ADC),which is much greater than the chip rate T_(C) of the bi-phasemodulating signal.

The sensing method of the sensor unit 20 is operative to follow a RandomDemodulator as readily understood by those skilled in the art. However,to complete a communication link providing both forward error correction(FEC) and information assurance (IA), the typical random demodulatorapproach is augmented with compressively encoded symbols into the datastream for the express purpose of correcting channel errors anddetecting that the message was generated by an authorized user. Theselast two features are unique aspects of the illustrated sensing andrecovery system 10. The FEC and IA data are also available to be encodedinto the transmission stream.

A data integrity module 36 adds data integrity check symbols to thecompressed sensed signal. The data integrity module 36 generates theintegrity check symbols based on a matrix 37, wherein the matrix may berandomly generated based on a received IA key or seed. This allows thereceiving unit 60 to know if the message is authentic. Otherwise, thereceiving unit 60 may be susceptible to being spoofed. The sensing unit20 also includes a forward error correction (FEC) module 38 to addparity check symbols to the compressed sensed signal. This allowscorrection by the recovery unit 60 for any errors that may be introducedin the communications channel due to noise.

In this non-limiting example, more particularly, the authenticationsymbols are appended to the symbols of the data stream. As anon-limiting example, when the authentication symbols are consideredalready in the digital domain, a standard compressive sensing technique(i.e., a random matrix) may be applied to the IA data to create thecompressively sensed symbols in the data frame. Given the IA data is ofa different format than the analog source data, the sensing unit 20 isoperative to encode the authentication word using an ancillarycompressive sensing circuit.

Further, the IA symbols are programmed in the sensing unit 20 using ameans suitable per the end application (e.g., hardwired, commanded,etc.) a-priori to data transmission. The IA source data is selected suchthat it will be sparse in the same domain as the source data signal tobe transmitted. The advantage of this encoding method is that a commonrecovery circuit for the data stream and IA symbols may be used in therecovery unit 60. Otherwise, separate recovery systems for the data andIA symbols can be used, but at the cost of additional complexity in therecovery unit 60.

The IA encoding mechanism and the position of the IA samples in thecompressively sensed data stream are both known to the recovery unit 60.After the IA “compressed” IA symbols are appended to the data payload,FEC is applied to complete the symbol block. The FEC is applied usingany suitable traditional coding approach (e.g., block, convolution,turbo) so long as the recovery unit 60 is apprised of the location ofthe FEC symbols. The FEC symbols need not be compressively encoded aschannel errors may cause catastrophic errors in signal recovery.However, using compressive techniques to provide FEC is not excluded.

In the recovery unit 60, the signal reconstruction unit 62 will form amatrix V which maps the sparse data coefficients (i.e., the data and IAsymbol stream) to the compressed measurements. That is, the receivedsignal y=Vα in vector form, where the original signal x can berepresented in vector form based on a basis Ψ and expansion coefficientsα associated therewith. In other words, x=Ψα in vector form. Theelements of the Ψ matrix depend on the bi-phase sequence and thesparsification transform in the sensing unit 20. The resultingconstrained optimization problem is expressed as minimizing the L−1 normof the α subject to the constraint of y=ΦΨα.

Solving the resulting under-constrained equation using an L−1 normproduces a set of sparse coefficients. The resulting sparse coefficientsare then converted back to signal samples and IA samples, using anassumed sparsification transform, that would have been acquired had theentire process been executed at the Nyquist rate.

The sensing unit 20 can further randomize its operation to furtherinsure that unauthorized terminals cannot intercept or transmitcompatible data streams. The additional randomization includesrandomization of the seed values generating the bi-phase modulationsignal and the seed generating the IA compressed symbol stream. Therandomization can be applied as a randomly selected start value and/orduring a transmission randomly resetting the random generators with newseed values.

An inherent benefit of this is that the transmitted compressed sensedsignal now has an even lower probability of intercept (LPI). Since theseed controls the exact nature of the bi-phase sequence, the more oftenthe seeds are changed, then the more difficult it is to intercept thetransmitted compressed sensed signal.

However, offsetting some of the benefit of additional randomization forsignal protection, is the increased complexity in the recovery unit 60,where the seed changes need to be likewise synchronized to alterationsin the sensing unit 20. There are many methods (e.g., GPS-based, orderwire, etc.) that can be used as readily appreciated by those skilled inthe art.

The overall system 10 is operative to accept a bandlimited continuoustime analog signal in the sensing unit 20. In order for the compressiveparadigm to be applicable, the signal is assumed to be sparse in somebasis Ψ. Depending on the type of signal, certain basis are bettersuited for certain types of signals, as also readily understood by thoseskilled in the art. Many naturally occurring and man-made signals areknown to have a sparse representation (i.e., a few non-zero weights inthe linear combination of basis vectors) in some transform basis (e.g.,Fourier, DCT, wavelets, etc.). The sensing unit 20 does not need to knowthe basis nor compute with it. However, the recovery unit 60 needs toknow the basis, or as a minimum, assume a basis that sufficientlysparsely represents the signal.

A more detailed block diagram of the recovery unit 60 will now bediscussed in reference to FIG. 3. The received encoded data is firstpassed to an error detection and correction module 68 that corrects forany errors as determined based on the parity check symbols added by theFEC module 38 in the sensing unit 20. Channel errors, if leftuncorrected, will likely make reconstruction of the data completelyincorrect.

The received encoded data after error correction is then passed to anintegrity check module 69 to determine if the sensing unit 20 isauthorized. The integrity check module 69 compares the data integritycheck symbols added by the data integrity module 36 in the sensing unit20. The integrity check module 69 generates the integrity check symbolsbased on a random sensing matrix 67, which is similar to those used instandard approaches to discrete signal compressive sensing, as readilyappreciated by those skilled in the art. The matrix 67 in the integritycheck module 69 and the matrix 37 in the data integrity module 36generate values based on the same IA key or seed that is known betweenthe two modules. If the data integrity check symbols added by the dataintegrity module 36 in the sensing unit 20 are verified by the recoveryunit 60, then the sensing unit 20 is authorized and the received encodeddata after error correction is passed to the signal reconstructioncircuit 62.

If the sensing unit 20 is not authorized, then the recovery unit 60ignores the received encoded data after error correction. If the sensingunit 20 is authorized, then the received encoded data after errorcorrection recovered data is passed to the signal reconstruction circuit62.

The recover unity 60 includes a signal reconstruction circuit 62 thatreceives the transmitted compressed sensed signal y, which is a sequenceof discrete time outputs. As noted above, the transmitted compressedsensed signal y=ΦΨα. The recovery unit 60 knows the sensing time varyingand “randomly chosen” matrix Φ. The recovery unit 60 also knows thebasis Ψ, for expressing the signal.

Alternatively, the recovery unit 60 may operate properly even if adifferent but suitable basis is used, as noted above. There are manysparsification transforms that may be used to represent the sensed data.Depending on the data being sensed, certain basis are better suited. Thedata at least needs to be sparse in an anticipated basis that isacceptable for the recovery unit 60, otherwise the process will notwork.

The signal reconstruction circuit 62 recovers the compressed sensedsignal using two steps. The first step is performed by the determinebasis coefficients section 64, which performs a constrained optimizationto solve for the basis coefficients α. The constrained optimization isbased on the L1-norm function. The second step is performed by applyingthe coefficients to the basis section 66, which applies the coefficientsα to the basis Ψ to determine x(t). The recovery unit 60 is able toreconstruct what the Nyquist samples would have been had they had beendigitally sampled with a conventional system. The recovered data ispassed to a data output module 65.

A flowchart 200 illustrating a method for sensing data will now bediscussed in reference to FIG. 4. From the start (Block 202), the methodcomprises generating a bandlimited continuous analog signal at Block204, and generating a modulated analog signal based upon modulation ofthe bandlimited continuous time analog signal at a modulating rate atleast equal to a Nyquist rate for the bandlimited continuous time analogsignal at Block 206. A compressed sensed signal is generated at Block208 based upon conversion of the modulated analog signal at a samplingrate less than the Nyquist rate. Authentication symbols are added to thecompressed sensed signal at Block 210 and error correction symbols areadded to the compressed sensed signal to protect the overall messagefrom the effects of channel errors at Block 212. The method furthercomprises transmitting the compressed sensed signal to a recovery unit60 at Block 214. The method ends at Block 216.

A more detailed explanation on compressive sensing will now be provided.Compressive sensing deals with the problem of acquiring an M×1discrete-time signal vector of samples, denoted as y and referred to ascompressed measurements, to represent a signal x(t) that is K-sparse orcompressible in some domain. We denote the sparisification transformemployed in the receiver as Ψ(t). Generally the sparisificationtransform is time non-adaptive, however, this possibility is notexcluded.

Compressive sensing has been traditionally applied to data that isalready discretized in some fashion (e.g., pixilated data, time series),and the compressive sensing concept is useful to reduce data storage ortransmission requirements. The mathematics are fairly straightforwardmatrix-vector equations. The discrete data domain serves as a good placeto introduce compressive sensing, and builds an insight for howcompressive sensing can be used to acquire analog signals at sub-Nyquistrates, and reconstructed with zero error (or as if the signals wereNyquist sampled to begin with). Some important characteristics affectingthe hardware design of this technique are that sampling is non-adaptiveand periodic. So the acquisition circuitry is somewhat agnostic to theinput signal.

A way to formulate (in discrete-time) to the compressed samples is thefollowing. Consider a discrete-time vector xε

^(N×1) and a random matrix Φε

^(M×N) and form the samples (measurement or observations) y as, y=Φxε

^(M×1). There is a limit as to how small M can be selected when K isfixed. K is influenced by the choice of the basis. A typical rule ofthumb, when using the L1-norm for recovery, is M˜0(2K log(N/M)). Whenthe representation of the vector x is expressed in the sparse basis weobtain, x=Ψαε

^(N×1).

Then by combining equations, the following is obtained: y==ΦΨα=Vαε

^(M×1). The original vector x can be recovered exactly by solving aconvex optimization using the L1-norm with equality constraints. Namely,min Σ_(k=1) ^(N)|α_(k)| such that y=Vαε

^(M×1). The original data is then recovered using x=Ψαε

^(N×1).

The recovery problem is ill-posed in the matrix-vector form. There arean infinity of solutions because there are more unknowns than equations.It is the fact that the signal is assumed sparse in a basis allows therecovery process to operate. If the signal is not sparse directly in thedomain of the samples that create the observations a sparsificationtransform must be found to implement the process. This line of thinkingwill be useful when the concepts are transferred to the compressivesensing of analog signals.

However, as a special case, if the data vector x is already sparse, sayin the discrete-time domain, it does not need to be made further sparse.It can be directly used in the above formulation without introducing thematrix Ψ at all. In this case the equations become y=Φxε

^(M×1) (data to observations). The sparse data is recovered using theknown random matrix Φ, using the same type of convex optimization usingthe L1-norm with equality constraints, which in this case is, minΣ_(n=1) ^(N)|x_(n)| such that y=Φxε

^(M×1). The optimization yields the original N×1 vector x directly fromthe M×1 measurement vector y.

Application to continuous time signals will now be discussed. Discretetime samples may be created from a continuous time signal as if it wereNyquist sampled, yet at a lower than Nyquist sample rate for thedigitizer. The first issue is how to convert the continuous time systeminto a form amenable to the matrix-vector formulation given in theprevious sections.

First, it is assumed that the analog signal has a finite informationrate, so then it is reasonable to assume that it can be represented by afinite number of parameters per unit time in some continuous basis.Namely the expansion

${x(t)} = {\sum\limits_{k = 1}^{N}{\alpha_{k}{\psi_{k}(t)}}}$

is valid. The choice of the basis functions is dictated by how fewcoefficients the user wishes to have as non-zero. A guideline is thatsparser representations are preferred. Note that since N coefficientsare allowed, in the “worst case” in the sense that the components cannotbe ignored, the basis functions could be a sequence of time-shifted sinefunctions or some other Nyquist pulse shape, and each coefficients arethe functional values at each (sampling) instant.

While each dictionary function may have a high bandwidth, the signalexpressed in the basis has relatively few degrees of freedom. Ideally,the signal could be sampled at some multiple level of the sparsityrather than twice the bandwidth as dictated by the Nyquist theorem.

The acquisition system has 3 parts: the random modulator, filtering anduniform sampling. The random modulator 24 uses a chipping sequence. Thesequence is as fast or faster than the Nyquist rate of the input signal.The purpose of the modulation is to spread the frequency content so thatis it not destroyed by the LPF 26. The spread signal is filtered, andsampled in an analog-to-digital converter 32 at a rate T<<Nyquist rate.

To recover the original signal as if it were originally Nyquist sampled,the observations are expressed in a matrix-vector formation suitable forthe L1-norm optimization, min Σ_(k=1) ^(N)|α_(k)| such that y=Vαε

^(M×1).

The matrix V is to be determined. To derive it, the following is noted:

y[m] = ∫x(τ)p_(c)(τ)h(mT − τ)τ${y\lbrack m\rbrack} = {\sum\limits_{k = 1}^{K}{\alpha_{k}{\int{{\psi_{k}(\tau)}{p_{c}(\tau)}{h\left( {{mT} - \tau} \right)}{\tau}}}}}$

To recover a matrix-vector relationship of y=Vα, the entries of V aredefined as follows: V_(m,n)=∫ψ_(k)(τ)p_(c)(τ)h(mT−τ)dτ. All threefunctions are known to the recovery unit 60.

The sensing matrix Φ will now be discussed. In a discrete system thesensing matrix Φ may ideally be Gaussian, because many of the provableresults are for random matrices with Gaussian entries. However, as apractical matter, a random matrix where the rows are outcomes of arepeated Bernoulli trial yield results commensurate with the Gaussiantheory. This observation on discrete-time system processing led to theinclusion of the chipping sequence in the continuous time version,notwithstanding the fact that it is straightforward to build a bi-phasemodulator 24. Chaotic versions have also been shown to provide provablygood results. Also, the nature of the sampling matrix Φ must be selectedso that it is noncoherent with the sparsification transform ψ.

There is a decrease in SNR in the recovery stage when there is noise inthe system. Given an input SNR in the original (uncompressed signal) andthe compressively sensed measurements, a 3 dB/octave penalty is paid.This is due to the wideband noise folding into a narrow band output fromthe analog-to-digital 32.

When the source signal has noise induced by some mechanism (e.g.,external coupling, circuitry preceding random demodulator, randomdemodulator components, quantization noise, etc.) the L1-norm recoveryprocess (i.e., L1-norm recovery with equality constraints) is modified.In this case, a quadratically constrained L1-norm is used, as follows:

min Σ_(k=) ^(N)|α_(k)| such that ∥y−Vα∥ ₂ ²<ε

In this case, the recovery is again a 2-step process, just as in theabove sub-Nyquist application. First, solve the constrained optimizationproblem using y=Vα as the constraint equation. Then, transform the αvector into the desired source signal s using the relations s=ψα. Alsoin this case, since ψ≠I a dictionary entry that induces sparseness mustalso be remembered by the source compression system, as well as therandom sensing matrix Φ, since together they form V.

Another aspect is to apply compressive sensing to a data system tosecurely transmit and retrieve source data files with respect to aremote data storage facility. As will be explained in greater detailbelow, random generation of the sensing matrix may advantageouslyprovide an unlimited number of 1-time encryption pads when generatingcompressed sensed data files. In effect, the sensing matrix is beingused as an encryption key.

Referring now to FIG. 5, a data handling system 300 includes acompressive sensing circuit 310 configured to receive a source datefile. The data handling system 300 may generate the source data file, ormay receive the source data file from an external source.

The compressive sensing circuit 310 includes a sparseness module 312 anda measurement module 318. The sparseness module 312 generates a sparsesource data file by inducing sparseness into the source data file. Aswill be explained in greater detail below, sparseness may be introducedbased on direct embedding or a sparsification transform.

The measurement module 318 generates the sparse source data file (i.e.,the compressive samples) based on a sensing matrix 317 to generate acompressed sensed source data file. The compressed sensed source datafile is to be transmitted to a remote data storage facility 330 forstorage. The remote data storage facility 330 is also referred to ascloud storage since the compressed sensed source data file is sent overthe Internet 332 via interface 324 to the remote data storage facility330.

A recovery unit 340, at some point later in time, retrieves thecompressed sensed data file from the remote data storage facility 330.The recovery unit 340 generates the source data file based onapplication of the same sensing matrix 317 used by the measurementmodule 318. A data reconstruction module 342 within the recovery unit340 reconstructs the same sensing matrix. The source data file cannot berecovered without using the same sensing matrix. In effect, the sensingmatrix is being used as an encryption key.

The recovery unit 340 is also configured to perform a trial recovery ofthe compressed sensed source data file prior to being transmitted to theremote data storage facility 330 for storage, and if the trial recoveryis successful, then perform the transmitting. The measurement module 318is configured to re-generate the compressed sensed source data file fromthe sparse source data file and based on a new sensing matrix if thetrial recovery is unsuccessful. The recovery unit 340 is configured toperform a trial recovery on the re-generated compressed sensed sourcedata file prior to being transmitted.

The trial recover feature of the data handling system 300 guards againstthe possibility that the random entries chosen to encode the data in thesensing matrix 317 lack the proper structure to successfully recover thedata. The trial recovery feature circumvents the issue by repeatedlygenerating sensing matrices 317 and applying them to the source datauntil the trial recovery is successful.

A more detailed block diagram of the data handling system 300 will nowbe discussed in reference to FIG. 6. The illustrated data handlingsystem 300 includes a buffer or memory 306 to initially receive thesource data file. A compression circuit 308 compresses the source datafile by removing redundancy therein. The compression circuit 308 is notlimited to any particular compression algorithm. Instead, the choice ofthe compression algorithm may be dictated by the nature of the sourcedata file, as readily understood by those skilled in the art.

The sparseness module 312 induces sparseness into the compressed sourcedata file. The sparseness may be induced by direct embedding or by asparsification transform.

Direct embedding will be discussed first. For discussion purposes, it isassumed that the output of the compression circuit 308 is a vector c,where c has dimensions B×1, and B denotes the number of bits in thesource data file. Typically, the compressed file is not sparse ascompression (e.g., JPEG, MPEG, ZIP, etc.) seeks to make the file assmall as possible. It is possible to expand the file size (i.e., inducesparseness) by embedding the file (the vector c) into a much larger allzero vector resulting in a vector s of dimensions kB×1, where k is asuitably large multiplier (e.g., typically 5-10 or more).

One way to embed the data vector c into the required sparse vector s isthe following: s=[c′,0,0,0,0,0, . . . 0]′. Of course, any method fordispersing the data is readily acceptable. Since a sparse vector isavailable, the transformation from source data into a sparse vector,namely ψ, is the identity matrix (in this case). So with ψ=I, therelation y=Vα=φψα=Φs as before in the sub-Nyquist application. Thedimensions of Φ are M1×kB.

Since ψ=I, the constrained L1 recovery can be cast directly using theconstraint equation y=Vα=φα=Φs. In other words, the sparserepresentation α is the same as the data s out of the induce sparsenessmodule 312 (which as explained above is by construction sparse).

The main advantage of this technique over the other approach of using asparsification transform is that no sparsification transform isnecessary. However, a penalty may be paid in the sense that the data ysent to the cloud may be larger as compared to using the sparsificationtransform. When the recovery is performed, the data recovered (in theknown embedded locations) is the data needed by the recovery unit 340for information recovery. In other words, the sparse recovery is onestep since ψ=I.

In the sparsification transform approach to induce sparseness, the datavector from the compression circuit 308, denoted above as the vector c,is used again. This time, the data vector c is used directly and testedagainst various dictionaries (i.e., sets of ψ's or equivalentlydifferent ψ matrices) to see if the combination of data and a dictionaryentry admit a sparse set of coefficents, namely the α vector. This isrepresented mathematically as c=s=ψα, where c=s because the sourcesignal to be compressively sensed is the vector c.

If there is no sparse representation of the source data c, thencompressive sensing will fail. But assuming that either a sparsetransform is known a-priori or can be learned online from the data, thenthe vector c is passed through the sparseness module 312, and themeasurements y to be archived in the cloud are formed as y=Φs=Φc=Φψα=Vα.Namely, the vector c (B×1) is taken and the random sensing matrix (M2×B,M2<<B) is applied to the data, wherein the result y is stored in thecloud.

As a result of the reduction in size of the source vector and thepossible efficient representation induced by a suitably chosen ψ matrix,M2 may be much smaller than M1, yielding improved output memoryefficiency. The penalty paid for such an efficiency is that a suitablesparsification transform must be known a-priori or learned.

In this case, the recovery is again a 2-step process, just as in theabove sub-Nyquist application. First, solve the constrained optimizationproblem using y=Vα as the constraint equation. Then, transform the αvector into the desired source signal s using the relation s=ψα. Also inthis case, since ψ≠I a dictionary entry that induces sparseness mustalso be remembered by the source compression system, as well as therandom sensing matrix Φ, since together they form V.

Just as in the sub-Nyquist application, it is permissible that differentψ matrices may be better than others as a signal's time record evolves.There is no restriction that one ψ matrix be used for an entire file tobe stored. All that is required is that the recovery unit 340 know whento switch among the dictionary choices and what those choices are.

After the sparseness module 312, the compressive sensing circuitincludes a data integrity module 314 and a forward error correction(FEC) module 316 to further enhance the confidentiality and integrity ofthe data. As discussed above, the data integrity module 314 adds dataintegrity check symbols to the source data file. The data integritymodule 314 generates the integrity check symbols based on a matrix 315,wherein the matrix may be randomly generated based on a received IA keyor seed. This allows the recovery unit 340 to know if the retrievedsource data file is authentic and has not been altered or tampered. Asalso discussed above, the FEC module 316 adds parity check symbols tothe source data file. This allows correction by the recovery unit 340for any errors that may be introduced.

The measurement module 318 is configured to randomly generate eachsensing matrix 317 based on a polynomial and a seed, as readilyunderstood by those skilled in the art. In one embodiment, themeasurement matrix 318 includes a chaotic generator to chaoticallygenerate the seed for each respective sensing matrix 317.

A memory 320 is coupled to the measurement module 318 and the recoveryunit 340 to store the polynomial and the seed for each respectivesensing matrix 317. The recovery unit 340 reconstructs the same sensingmatrix 317 based on the stored polynomial and seed to generate thesource data file. The measurement module 318 also stores a time stamp inthe memory 320 corresponding to when the sensing matrix 317 wasgenerated. The time stamp is used to identify the seed used to generatethe sensing matrix 317. The recovery unit 340 retrieve the seed from thememory 320 to generate the sensing matrix 317 based on the time stampassociated therewith.

The recovered source data file is passed to an error detection andcorrection module 352. The error detection and correction module 352corrects for any errors as determined based on the parity check symbolsadded by the FEC module 316 in the compressive sensing unit 310.

An integrity check module 350 is then used to determine if the sourcedata file is authentic. The integrity check module 350 compares the dataintegrity check symbols added by the data integrity module 314 in thecompressive sensing unit 310. The integrity check module 350 generatesthe integrity check symbols based on a matrix 351. The matrix 351 in theintegrity check module 350 and the matrix 315 in the data integritymodule 314 generate values based on the same IA key or seed that isknown between the two modules.

If the data integrity check symbols added by the data integrity module314 in the compressive sensing unit 310 are verified by the recoveryunit integrity check module 350, then the recovered source data file isaccepted. The accepted source data files are then passed to the recoveryunit 340. Otherwise, the retrieved source data file is ignored, whichmeans that the source data file has been altered or tampered.

The data reconstruction circuit 342 within the recovery unit 340recovers the source data file from the compressed sensed source datafile retrieved from the remote data storage facility 330 using two stepsand based on the sensing matrix 317. The first step is performed by thedetermined basis coefficients module 346, which performs a constrainedoptimization to solve for the basis coefficients α. The constrainedoptimization is based on the L1-norm function. The second step isperformed by the apply coefficients to basis module 348, which appliesthe coefficients α to the basis ψ to determine x(t), which is equal toψα. The recovery unit 340 is able to reconstruct the compressed sourcedata file based on the same sensing matrix 317 used by the compressivesensing unit 310.

A decompression circuit 360 corresponding to the compression circuit 308decompresses the reconstructed source data file which may then be storedin the memory 306.

A flowchart 400 illustrating a method for operating a data handlingsystem 300 will now be discussed in reference to FIG. 7. From the start(Block 402), the method comprises receiving a source date file at Block404, and generating a sparse source data file by inducing sparsenessinto the source data file at Block 406. A compressed sensed source datafile is generated at Block 408 from the sparse source data file andbased on a sensing matrix.

A mapping and a respective seed used in generating the sensing matrixare stored in a memory 320 at Block 410. The compressed sensed sourcedata file is transmitted to a remote data storage facility 330 forstorage at Block 412. The compressed sensed source data file is laterretrieved from the remote data storage facility 330 at Block 414. Themethod further comprises at Block 416 retrieving the stored mapping andthe respective seed based on the time stamp associated therewith toreconstruct the sensing matrix. The source data file is recovered fromthe retrieved compressed sensed source data file and based upon thereconstructed sensing matrix at Block 418. The method ends at Block 420.

Many modifications and other embodiments of the invention will come tothe mind of one skilled in the art having the benefit of the teachingspresented in the foregoing descriptions and the associated drawings.Therefore, it is understood that the invention is not to be limited tothe specific embodiments disclosed, and that modifications andembodiments are intended to be included within the scope of the appendedclaims.

That which is claimed is:
 1. A sensing and recovery system comprising: asensing unit comprising a sensor configured to generate a bandlimitedcontinuous time analog signal, a modulator coupled to said sensor andconfigured to generate a modulated analog signal based upon modulationof the bandlimited continuous time analog signal at a modulating rate atleast equal to a Nyquist rate for the bandlimited continuous time analogsignal, and a compressive sensing circuit coupled to said modulator andconfigured to generate a compressed sensed signal based upon conversionof the modulated analog signal at a sampling rate less than the Nyquistrate; and a recovery unit coupled to said sensing unit and configured torecover the bandlimited continuous time analog signal from thecompressed sensed signal.
 2. The system according to claim 1 whereinsaid compressive sensing circuit comprises an analog-to-digitalconverter.
 3. The system according to claim 1 wherein said sensing unitand said recovery unit are configured to be synchronized.
 4. The systemaccording to claim 1 wherein said recovery unit is configured to recoverthe bandlimited continuous time analog signal based on a basis function.5. The system according to claim 4 wherein said recovery unit isconfigured to determine basis coefficients from the compressed sensedsignal, and to apply the basis coefficients to the basis function torecover the bandlimited continuous time analog signal.
 6. The systemaccording to claim 1 wherein said sensing unit further comprises aforward error correction (FEC) module configured to add error correctionsymbols to the compressed sensed signal.
 7. The system according toclaim 6 wherein said receiving unit further comprises an error detectionand correction module configured to correct for errors in the compressedsensed signal based on the error correction symbols added by said FECmodule.
 8. The system according to claim 1 wherein said sensing unitfurther comprises a data integrity module configured to addauthentication symbols to the compressed sensed signal.
 9. The systemaccording to claim 8 wherein said data integrity module is configured togenerate the authentication symbols based on a randomly generatedmatrix.
 10. The system according to claim 9 wherein said receiving unitfurther comprises an integrity check module configured to authenticatesaid sensing unit based on the authentication symbols added by said dataintegrity module.
 11. The system according to claim 10 wherein saidintegrity check module is configured to generate the authenticationsymbols based on the same randomly generated matrix.
 12. The systemaccording to claim 1 wherein said modulator comprises a bi-phasemodulator.
 13. A sensing unit comprising: a sensor configured togenerate a bandlimited continuous time analog signal; a modulatorcoupled to said sensor and configured to generate a modulated analogsignal based upon modulation of the bandlimited continuous time analogsignal at a modulating rate at least equal to a Nyquist rate for thebandlimited continuous time analog signal; and a compressive sensingcircuit coupled to said modulator and configured to generate acompressed sensed signal based upon conversion of the modulated analogsignal at a sampling rate less than the Nyquist rate.
 14. The sensingunit according to claim 13 wherein said compressive sensing circuitcomprises an analog-to-digital converter.
 15. The sensing unit accordingto claim 13 further comprising a forward error correction (FEC) moduleconfigured to add error correction symbols to the compressed sensedsignal.
 16. The sensing unit according to claim 13 further comprising adata integrity module configured to add authentication symbols to thecompressed sensed signal.
 17. The sensing unit according to claim 16wherein said data integrity module is configured to generate theauthentication symbols based on a randomly generated matrix.
 18. Thesensing unit according to claim 13 wherein said modulator comprises abi-phase modulator.
 19. A method for sensing data comprising: generatinga bandlimited continuous analog signal; generating a modulated analogsignal based upon modulation of the bandlimited continuous time analogsignal at a modulating rate at least equal to a Nyquist rate for thebandlimited continuous time analog signal; and generating a compressedsensed signal based upon conversion of the modulated analog signal at asampling rate less than the Nyquist rate.
 20. The method according toclaim 19 further comprising transmitting the compressed sensed signal toa recovery unit.
 21. The method according to claim 19 wherein thecompressed sensed signal is generated using an analog-to-digitalconverter.
 22. The method according to claim 19 further comprisingadding error correction symbols to the compressed sensed signal.
 23. Themethod according to claim 19 further comprising adding authenticationsymbols to the compressed sensed signal.